Deconstructing Hopf Spaces

We characterize Hopf spaces with finitely generated cohomology as algebra over the Steenrod algebra. We “deconstruct” the original space into an Hspace Y with finite mod p cohomology and a finite number of p-torsion EilenbergMac Lane spaces. One reconstructs X from Y by taking extensions by principal H-fibrations. We give a precise description of homotopy commutative H-spaces in this setting and give a criterion to recognize connected covers of H-spaces with finite mod p cohomology. The key observation is that the module of indecomposables QH∗(X ;Fp) lies in some stage of the Krull filtration of the category of unstable modules over the Steenrod algebra. We compare this algebraic condition with a topological one, namely that some iterated loop space of X is BZ/p-local.